Shock polars for non-polytropic compressible potential flow

Figure 1.  Supersonic flow onto a slender wedge. The strong-type shock does not usually appear

Figure 2.  Shock polar (full Euler, polytropic $ \gamma = 7/5 $, $ M_0 = 3 $)

Figure 3.  $ v^x, v^y $ plane shock polar for polytropic potential flow, $ \gamma = 5/3 $ and $ M_0 = 3.5 $

Figure 4.  $ j,\theta $ plane shock polar for polytropic potential flow, $ \gamma = 5/3 $ and $ M_0 = 3.5 $

Figure 5.  Mass flux reaches a maximum at critical speed ($ v = c $; the overused term "critical" means "sonic" in this context, in contrast to the shock polar context)

Figure 6.  $ h $ is strictly convex in $ -1/2 \varrho^2 $. Tangents to its graph have slope $ ( \varrho c)^2 $, the chord from up- to downstream state represents $ ( j^n)^2 $. The Lax condition $ \varrho_0c_0< j^n_0 = j^n< \varrho c $ is obvious here

Figure 7.  Shock-velocity angles $ \beta $ and turning angle $ \theta $ in relation to shock, shock normal, mass flux and velocity vectors; $ \varrho_0 = 1 $ chosen to make $ {\mathbf{v}}_0 = {\mathbf{j}}_0 $ for the diagram. Decreasing $ \beta_0 $ while holding $ \theta $ fixed increases both $ v $ and $ j $, which requires $ M<1 $, explaining why critical-type shocks are transonic since their $ \theta $ is fixed to first order